The projection technique of Sanchez-Portal et al.
[48] have been implemented. The eigenstates
obtained from PW calculations when
sampling at a given wavevector 
are projected onto Bloch
functions formed from a localised basis set 
. In this case, the natural choice of basis set is the set of
atomic pseudo-orbitals generated from the pseudopotentials used in the
calculation. However, care must be taken when performing the
projection because this basis set is neither orthonormal nor complete.
The overlap matrix of the localised basis set, 
, is
defined :
These overlaps are computed in reciprocal space, as the imposition of periodic boundary conditions implies that the orbitals need only be evaluated on a discrete reciprocal space grid. The overlaps between orbitals on different atomic sites can be calculated on the same grid with the application of a phase factor. The representations of the atomic pseudo-orbitals on the discrete reciprocal space grid may be calculated by Fourier transformation of the real space wavefunctions generated during construction of the pseudopotentials, or can be generated using the CASTEP code.
The overlaps between the plane wave states and the basis functions,
are also calculated in reciprocal space, as this is the natural representation of the plane wave states.
The quality of the projection may be assessed by the calculation of a spilling parameter,
where 
is the number of PW eigenstates, 
are the
weights associated with the calculated points in the Brillouin
zone and 
is the projection operator of Bloch functions with
wave vector generated by the atomic basis.
where 
are the duals of the atomic basis
states such that
The spilling parameter varies between one, in the case that the atomic
basis set is orthogonal to the PW eigenstates and zero, when the
projected wavefunctions perfectly represent the PW states. Unlike the
procedure of Sanchez-Portal et al. [48], the
basis functions are not scaled in order to optimise 
. This
gives spilling parameters in the region of 
which are
believed to be adequate for the purposes of this work.
The density operator may be defined,
where 
are the occupancies of the PW states, 
are the projected eigenstates, 
, and 
are the
duals of these states. From this, the density matrix for the atomic
basis set may be calculated as follows:
may be calculated in a time proportional to the cube
of the system size, the same order of scaling as the original PW
calculations, but the total time is smaller as there is no need to
carry out iterations to achieve self-consistency.
The density matrix and the overlap matrix are sufficient to perform population analysis of the
electronic distribution. In Mulliken analysis [43] the
charge associated with a given atom A is given by
and the overlap population between two atoms A and B is
In Löwdin analysis [44] these quantities are given by
and
In Equations (
) and () the matrix
is calculated as a Taylor expansion of
where 
.
Details of the implementation of these methods may be found in
Appendix .
